K.B. Wolf, Canonical transforms. IV. Hyperbolic transforms - UNAM

Canonical transforms. IV. Hyperbolic transforms: Continuous series of SL{2,R) representations Kurt Bernardo WoW) Centre de Recherches Mathematiques. Universite de Montreal. Montreal. Quebec. Canada

(Received 5 January 1979; accepted for publication 23 April 1979) We consider the sl(2,R) Lie algebra of second-order differential operators given by the Schrodinger Hamiltonians of the harmonic, repulsive, and free particle, all with a strong centripedal core placing them in the C ~ continuous series of representations. The corresponding SL(2,R) Lie group is shown to be a group of integral transforms acting on a (two-component) space of square-integrable functions, with an integral (matrix) kernel involving Hankel and Macdonald functions. The subgroup bases for irreducible representations consist of Whittaker, power, Hankel, and Macdonald functions. We construct the operator which intertwines this realization of SL(2.R) with the more familiar Bargmann realization on functions on the unit circle. This operator implements the canonical transformation of the above Schrodinger systems to action and angle variables.

q =:"/1 4""

1. INTRODUCTION

The program to explore the role of canonical transformations in quantum mechanics followed by Moshinsky and collaborators!.' has lead to advances and applications in three related fields: (a) It has given a better understanding of the dynamical groups (as opposed to dynamical or similarity algebras) for quantum systems and partial differential equations,'" (b) it has brought a significant unification into the theory of integral transforms,s-7 and (c) it has complemented the study of the three-dimensional Lorentz group generated by algebras of second-order differential operators. 8- 10 In this article, the fourth ofa series, 5.6.11 we would like to explore the following territory: Consider the three operators

J1 =

~(_ ~_ ~_p2),

(1.1 a)

J2 =

-

I) "2i (d p dp + "2 '

(l.lb)

J/}

4"1 ( -

=

dp2

4

p2

d 2 f.1 2) 1 dp2 - p2 +p , f.1> 4'

-i.JI.l'

[.JJ 2,Jd=iJJ

"

k (1 -- k) = ~(1 4

+ A 2) > ~4'

(l.3b)

k = !..(l + iA). A 2 = f.l - :..4 > D. (1.3c) 2 i.e., this set of operators belongs to the continuous or principal series of representations Cq as defined by Bargmann. 12 In the proper function domain-so that Eqs. (1.1) be self-adjoint-their spectra will have no lower bound.!] The potential singularity at the origin is indicative of the rather delicate domain problems we would find should we meet the problem starting from the algebra. This has been emphasized by Mukunda and Radhakrishnan,lo who also considered this realization. I n Sec. 2, we shall embed the sl(2,R ) algebra (1.1) as a subalgebra ofsp(4,R), reduced with respect to a "hyperbolic" suba\gebra soC 1,1) a! sl(2,R ). This chain is distinct from the "radial" so(2) a! sl(2,R ) chain considered in Refs. 3, 6 (Appendix B), and 14. The parameterization of the plane iII hyperbolic coordinate will lead to a two-component space YM.9i +) = y'2(.9? + ) + y 2 (&fi ) of square-integrable functions on the half-line, as the appropriate domain for Eqs. (1.1), carrying both the C ~ and C ~/2 representations. In Sec. 3 we consider the Lie group SL(2,R )=Sp(2,R ) generated by Eqs. (1.1), associated with the corresponding group of matrices through

(Uc)

which form an sl(2,R )=sp(2,R )=so(2, I) Lie algebra, with the well-known commutation relations

[Jl 1,J/2]=

+~ = 16

[J 3 ,.N 1 ]=i.JJ 2 • 0·2)

Among the algebra elements we have the Schrodinger Hamiltonians corresponding to a strongly attractive centripedal well (J 1 +.IT 3)' and similarly welled harmonic (2J 3 ) and repulsive (2J 1) oscillators. The algebra (l.I) constitutes the dynamical algebra for these systems. On calculating the value of the Casimir invariant of Eqs. (1.1), we find (1.3a) Q = J~ + J~ - J~ = q ll,

. ( cosh(a/2) exp(laJ 1) : _ sinh(a/Z) exp(i/3J 2 )

:

.

(exp( - /3 /2) 0

(COS(Y/Z)

exp(lyJ 3): sin(y/2)

I

- Sinh(a/2») cosh(a/2) ,

exp~ /2»).

- sin(y/2»). cos(y/2)

(1.4a) (l.4b) (l.4c)

whose adjoint action of the algebra-which is independent of the realization-is given by bd-ac ad

+ bc

_ bd - ac

!(a !(a

2 _

2

2 2_

b +c d -- cd - ab

2»)~1)

+ b 2 + c + d 2) 2

J2

(1.5)

•

.1

sabbatical leave from lIMAS, Universidad Nacional Aut6noma de Mexico. Apdo. Postal 20-726, Mexico 20, D.F.

alO n

680

J. Math. Phys. 21(4), April 1980

0022-2488/801040680-09$1.00

© 1980 American Institute of Physics

680

Downloaded 29 Jun 2011 to 132.248.33.126. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions

This group of automorphisms of the algebra will induce a corresponding group SL(2,R ) of integral transforms of x;( (~ +). In the first paper of this series,5 the algebra whose group ofautomorphisms was studied was the Heisenberg-WeyJ algebra of quantum mechanics. The group turned out to be, as here, SL(2,R ), but the integral transform carried the osciIIator (or metaplectic) representation D ;j4 -i- D 3~4' In the second paper6 it was the sl(2,R ) algebra-as here-which provided the "quantum mechanics" out of which we built the group of automorphisms (1.5) carrying the discrete D / series of representations. The integral transform kernel consisted of a Gaussian times a Bessel function. Here, it will involve Gaussian functions times Hankel and Macdonald functions of imaginary index. In contradistinction with the previous cases/.f' this integral transform group does not allow a complex extension in the group parameters to a unitary semigroup of transforms. In Sec. 4 we build the intertwining operator (i.e., the quantum mechanical canonical transform to action-and-angle variables) between the realization (1.1) ofsl(2,R) and the well-known Bargmann realization (~ofthe algebra in terms offirstorder differential operators on the circle 51:

JJ~

=

i e - i(COS - i, (0 < k < 1) is particularly troublesome, since various choices of boundary conditions 18 lead to representations which may belong to the lower-bounded discrete series ''lor to the unbounded supplementary series-a problem still to be solved for the algebra (1.1 )-which are not quite apparent in the formal expressions in Eqs. (1.1), and invisible in the classical Poisson-bracket construct. In establishing our results from the point of view of groups of integral transforms, we hope to settle some of the uncertainties which may arise in the algebraic approach to canonical transformations in quantum mechanics. FinalIy in Sec. 5 we outline some applications and offer some concluding remarks.

2. THE CHAIN sp(4,R)=>so(1,1) Ell sl(2,R) AND HYPERBOLIC COORDINATES

We consider the usual quantum mechanical operators of position and momentum in two dimensions [Qm f(q) = qm f(q) and P mf(q) = - iaj(q)/Jqm' m = 1,2] and out of these we build the symmetrized quadratic expressions Q", Qn' Pm P n , Qm ,IE\ J + . These ten operators span under Lie commutation the four-dimensional real symplectic algebra sp(4,R ), isomorphic to the pseudo-orthogonal algebra so(3,2). Let us denote the latter's generators in the Cartesian basis by

H

M'2 =~(QIP2 -Q2 P ]),M D = -!(p]PZ +Q]Q2)' ThH]4 = -HQ,P2 +Q2P]), M I5 = -!(p]P z -Q]Q2)'

MZ:l =

HP~

- P~ + Qi - QD, M24 = ~(Q] PI - Q2PZ)' (2.1)

!(Pi - P~ - Qi + QD, = -l(Pi +P~ -Qi -Q~), = l(q2)[where 1Ji~(q) = ( -1) nlJi~( - q) are the simple harmonic oscillator wavefunctions], and its spectrum will be given by m = !(n l - nz), nl,n z = 0,1,2, .... This set of functions will thus constitute a basis for the two continuous series representations ofsl(2,R): C~ spanned by the subset with n I + n 2 Kurt Bernardo Wolf

681

Downloaded 29 Jun 2011 to 132.248.33.126. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions

even [so that m is integer 1/1"1"' ( - q) = 1/1"1"' (q)], and C!/2 by the subset with n, + n 2 odd [m in half-integer and 1/1"1"' ( - q) = - 1/1"1"' (q)]. We shall now parametrize the plane in hyperbolic coordinates (p,,a), dividing it into two regions labeled by a as for qi - qi > 0:

a

= + 1, q, = p cosh, q2 = P sinh, p,ER;

for qT - q~ < 0:

a

= -

1, q,

(2.2a)

= p sinh, qz = p coshtP, (2.2b)

and disregard the cone qi - q~ = 0, as this is a submanifold of lower dimension. The elementsf(q) of the space offunctions Jf\.'3P 2) on the plane will be correspondingly representedby pairsoffunctionsfu(p,tP), a = ± I, elements ofa space Jfi (.'3P) -+- Jf2_, (.'3P) which can be arranged as a twocomponent vector column

f,(p,-( - p,tP,a),

lV

fl.,

=

- 00 14 = - i

00

1

25

= aw

00

dqz/(q"q2)*g(ql,q2)

~ ~,

- I12[

(2.5)

J2

- Jp2 - P

lV

00

JtP 2

[J

1

+ 2"

=

lV

,-'/2 [ J 2 - P -2 = 0023 = a';p - -

fl.)

= -

., -'/2 1¥J P Jp

fl.2

35

Lllz r'

X(~- ~)+p2]pl!2. JtP 2

J. Math. Phys., Vol. 21, No.4, April 1980

C!!2

E=.1(I-p),i.e.,E=0(1!2) for p= +1(-1). (2.7d)

BKo = - KoB ; BKk B,

k = 1,2,3,

(2.8b) (2.8c)

(2.6b)

(2.6c)

The operators (2.6) exhibit commutation relations analogous to Eq. (1.2). Acting on the column-vector function (2.3), the generators above will be represented by 2 X 2 diagonal matrices with operator elements, which for Eqs. (2.6a) and (2.6c) have opposite signs. The adjoint action of the group generated by Eqs. (2.6) on themselves can be verified to be formally identical to Eq. (1.5), as it should be, since the latter is a relation independent of the particular operator realization. For the a = - 1 components, we have a reversal of the signs of a and r in Eqs. (1.4), i.e., of band c in the elements of the 2 X 2 matrix realization in Eq. (1.5). This leaves the 3 X 3 matrix in Eq. (1.5) invariant. The subalgebras so(I,I) and sl(2,R) generated by Eqs.(2.5) and (2.6) are conjugate in sp(4,R ); the reduction to an irreducible subspace (irrep) of the former leads to a corresponding irrep of the latter. Since for sp(4,R ) itself we do not 682

i.e., it is the rotation-by-21T element of SL(2,R ) which commutes with the algebra sO(2,1)=sl(2,R) and which can be used to distinguish the vector and spinor constituent irrep's C~ and by demanding that lP' be diagonal. We use its eigenvalues p = ± 1 to distinguish the irrep spaces for C ~ through

B:(q, ,q2) ---->-(q" - q2),i.e., B(p,tP,a) --+(ap, - ,a), (2.8a) (2.6a)

Jp2

4

(2.7b)

Second, we have the inversion of the second Cartesian coordinate

- 2

X(~- ~) _p2]p'/2, 4

v=0,1,2,3,

(2.7c)

(2.4) in terms of the hyperbolic coordinates. Finally, the generators ofSO(l, 1) ® SL(2,R ) can be written as

Ko =

(2.7a) PlK.v=lK.)l',

af:+,f:oolpldP f:oodtPfa(P,tP)*gip,tP),

=

!!2,

(2.3)

The inner product in Jf2(.'3P 2) becomes (f,g)2 =

have a single irrep space but a direct sum of two-those with a basis with integer and with half-integer eigenvalues m under 00 45 or 0023 -the corresponding reduction of the sl(2,R ) generators will be the direct sum of two irrep'sC~ and C respectively. An irrep space for lK.owithin Eqs. (2.1) is provided by functions f!( p,tP ) = f!( p )exp(iAtP ), AE.'~. This will replace the operator - J 2/JtP 2 in Eqs. (2.6) by A 2 and bring the lK.k to within a similarity transformation (by p-ll2) of the forms (1. 1). In the following sections we shall be interested in certain discrete operations on the plane in Cartesian and hyperbolic coordinates which are, nevertheless, elements of the parent Sp(4,R ) group and which can be connected to the identity. These will be identified using the notation of Mukunda and Radhakrishnan 1o • First, we have the full space inversion

This element commutes with the sl(2,R ) algebra and with lP', but will intertwine the A and - A representations of so( I, I), and hence those of sl(2,R ). Its effect on the properly reduced irrep space C ~ will be to change the sign of the lower component of the E = ! function pair. Third, we have the element BlP', which will not interest us separately, and fourth, the operator A:(ql,q2)--+(q2,ql),i.e.,

AKj = KjA, j

=

A:(p,,a}--~(p,tP,

0,2; AKk = - KkA,

- a),

(2.9a)

k = 1,3,

(2.9b) A = B exp(i1TM,2)' This element does not commute with B (instead, AB = BlP' A), but it commutes with lP' and Ko and is thus representable as a unitary transformation in each C ~ irrep which reverses the sign of the K J eigenvalues. Its own eigenvalues (a = ± 1) will be used to classify the double-multiplicity Kl eigenfunctions. It is representable as a 0'1 Pauli matrix in the two-component function space (2.3). The A and B automorphisms are outer to SL(2,R ), while lP' is inner. Kurt Bernardo Wolf

682

Downloaded 29 Jun 2011 to 132.248.33.126. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions

and note that it will relate to Eq. (2.4) through

3. THE INTEGRAL TRANSFORM GROUP

The integral transform action of the Sp(4,R ) group generated by (2.1) on y2(&f 2 ) is known 20 .21. In particular, for the SL(2,R) subgroup generated by Eqs. (2.6), represented by the matrices 22 ,23

b0'3), dl

-=

(J'

3

it is

(3.1)

. , e--Iff

'CM (q,q')f(q'),

(3.2a)

where the integral kernel is, for b #0, CM (q,q') = (21Tlb J)

1

exp [i(a{q;2 _ q~2}

-2{q; ql - q~q2} + d

while for b

{qf -

qn)/2b ], (3.2b)

= 0, it is 1

CM(h,O)(q,q') =0- exp[ie(qi -qn/2a1

XO. Conversely,

i, (p,¢ ) = =

('f1 f;') (p,¢ )

I p7o+1

Joc ._=

dJ.f~/(p)

eXPCiA¢). (3.Sb)

-+

683

"'?i.

1

00

(3.7)

dA (f,g)p,A'

(3.8a)

(lK l ,lKz,lK3) = (o-JJ 1 ,JJ2 ,a.JI3 )'fP,A,

'fp-'lA = A'fP"\ 'fp,AB = 'fp,-,1.

(3.8c)

Equations (3,8a) only state that 'f p," indeed projects out eigens paces oflP and K.o, while Eqs. (3,8c) give relations which will be used later on. Equations (3.8b), finally, bring the three algebra generators (1.1) into the picture and, besides telling us that the special Sp(4,R) transform (3.2) leaves the (p,A ) subspace invariant, allows us to calculate the integral transform representing the operator C~ = 'fP''''CM which maps Yi, (&f+)P,,1 onto itself unitarily. Since the inner product (3.6) does not explicitly contain the labels p,A, we shall henceforth drop them from specifying the space ..2"i, (&f+). For functions/o-( p )EX'il (&f+), thus, the SL(2,R ) group generated by the operators .JIll, k = 1,2,3, acts as M

lu (p)

[C~fL (p)

----+

=

0"

f'..!c

I

i

oc

dp'C~~(T,a'

(p,p')!c" (p'),

l'" dp/~/(p)*g~/(p),

J_ Math. Phys., Vol. 21, No_ 4, April 1980

(:~.9a)

with the integral kernel C~~",(T' (p,p/) =pI/2('fP,,1CM,a,lT' ) (p,p')

(pp')I/2J~

=

'>0

dt,b[CM,(7,a' (p,,p;p',O)

+ pCM,(T,(T' (p,rp; -

p',O) ]exp( - iArp)

= GM,a,(T' (p,p')H~,~, (pp'lb ),

(3.9b)

where, on evaluating this expression from Eqs. (3.4) for b #0, we find it to be a product of a Gaussian factor GM,a,o' (p,p') = (21T1 b

I) -I

(pp') 112

X exp U(do-p2

+ ao-/p,2 )/2b ],

(3.10)

and a factor Hr::,~,(z) which contains the integration over t,b and which can be performed in terms of Hankel and Macdonald 25 functions, yielding 26 H~:1(z) =pHP!I, = 4p

i=

-1

(z)

drptrigp (z coshrp) cos (Arp)

= i1Tfpe·· A ,,-;2H )l)(z) - eA1T12H )]>(z)

J (3. 11 a)

Hf:AI (z) =pHP',{I,1 (z) =

4Pioc dt/ltrigp (z sinhrp)trigp (Arp) o

We define an inner product in the (p,A) subspace Yil (&f+)P,A = Yi (&1l+) y2 1 (,99+) as (f,g)p.A =

._

=pHP"'( -z) =HP'1 ,,{(z) 1.1 1'

1 _jpl-l/l 41T

I=

The properties of 'f p,A are such that 'fP,AIP = p'fP"\ 'fP,AK.o = 0 'fp,A,

CM (p,¢p;p',¢ ',0") = CM ( - p,¢,O'; - p',¢ ',a') = CM (p,t/J - t/J ' ,0-; p' ,0,0-' ) = (21Tlb I) -I exp[i(00-'p,2 -2pp'hYPO'.a' (¢' - ¢) + do-p2 )/2b ], (3.4a)

=

p~ I

(3.8b)

(01 0) 2q

1 41T

=

(JW,.JI~I,.JI~') = 'fP,A

ad - be = 1,

L,d

I(q) :[CMf](q) =

(f,gh

= 4(signz)2exp[ivtan( ¢ 12)]. , (4.7)

t

The generating function (4.1) is thus readily calculated from the integral as KP.A(",,p) =pl12Xpap-/2 ·A, (¢ )exp[icP X (p".t,O"p 212,0") J. '+' (T

(4.8)

In order to determine the phase function, consider the orthonormal Y 2(S() eigenbasis for .1l~ in C::

(4.2b)

where m is the integer for E = 0 (p = + 1) and half-integer for £ = 1/2 (p = - 1). The phase factors 7fn;A will be those of Bargmann 36 : rJk A = 1 = rJI/2(,A,

rf:;,A

= (-

(4.lOa)

1) m - € 1/2

m -

II

X

l~ E

p'),

+ co~'" )~ "'f' d¢

f/;/:;\¢) = (7fn;A] - 1(21T) - 112exp [i(m - £)¢ J, (4.9)

tarity of the transformation is guaranteed by the assumed Dirac orthonormality and completeness of the two eigenbases-including any similarity transformation as mentioned above-which, from Eq. (4.1) alone, implies


cP X(p,4,O"p 2/2,0") =-1 In(pI2)

(4.12)

where we have left a phase factor to be determined later on. Note that X~(P) is a two-component function which has only an upper component for v> 0 and only a lower one for v < O. As they stand, these functions may only involve the representation indices (p,4 ), if at alP1, in the phase factor

and declare the proper eigenfunctions of J~I corresponding to the eigenvalue m (integer or half-integer) to be

cPx(p,A,v,a).

spanning the C; irrep for sl(2,R ). On Eq. (4.13) we can ver-

685

J. Math. Phys., Vol. 21, No.4, April 1980

IJI ~u(p) = [7fn;A r (k X (pI2) -

+ O"m) ] l12

I

Wom . _ iA12(p2)

Kurt Bernardo Wolf

(4.13)

685

Downloaded 29 Jun 2011 to 132.248.33.126. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions

ify immediately that we have an eigenfunction of J~', as this operator acts as aJ 3 [see Eq. (Uc)] on the two a = + 1 and a = -1 components. Hance the eigenvalue is indeed m. Normalization under the inner product (3.6) carr be checked straightforwardly41. In order to support our claim that Eq. (4.12) is indeed an appropriate phase, we may verify that the action of J'± = Jl ± iJ~ on the simple functions tf/:/(

+ ae - Tn1/2) X [(tan (¢> /2» :;: 'I + a(tan (